The emergence of connected and automated vehicles (CAVs) has the potential to fundamentally improve mobility systems. It is widely recognized that a fully automated traffic system can greatly increase roadway capacity, reduce energy consumption, and improve road safety. However, full automation will be preceded by a long transitional period in which CAVs coexist with human-driven vehicles (HDVs) and public transit, making it important to leverage the advances of CAVs in mixed traffic environments. One proposed strategy is to repurpose existing dedicated bus lanes by allowing CAV access, but it is critical to control such access to avoid delaying buses or disrupting HDV traffic. This study proposes a bi-level control framework that integrates network-level routing with segment-level lane-changing control. At the network level, a dynamic routing strategy monitors real-time traffic flows on dedicated lanes and proactively reroutes CAVs before they enter edges where they may interfere with bus operations. At the segment level, the controller introduces a predictive lane-changing strategy that evaluates maneuvers based on anticipated downstream travel time, turning feasibility, and maneuver stability, while enforcing bus protection constraints through a monitoring window. The two control layers are coupled bidirectionally. The upper level supplies routes to guide lane-change feasibility, while executed lane changes are fed back to update upper-level flow estimates. Together, this interaction forms a unified and self-consistent control framework. The proposed framework is evaluated through microscopic traffic simulation in SUMO using a realistic urban corridor modeled on Van Ness Avenue in San Francisco. Results show that the integrated approach achieves 100% on-time arrival for buses while reducing average travel time for both CAVs and HDVs compared with uncontrolled scenarios. It also maintains stable lane-change behavior, ensuring safe operation. Sensitivity analysis further shows that the method remains effective under different parameter settings and monitoring horizons.